Multiple Impacts: A State Transition Diagram Approach

Transcription

1 Multiple Impacts: A State Transition Diagram Approach Yan-Bin Jia Department of Computer Science Iowa State University Ames, IA 5000, USA jia@iastate.edu Matthew T. Mason and Michael A. Erdmann School of Computer Science Carnegie Mellon University Pittsburgh, PA 523, USA {mason+,me}@cs.cmu.edu Abstract Impact happens when two or more bodies collide, generating very large impulsive forces in a very short period of time during which kinetic energy is first absorbed and then released after some loss. This paper introduces a state transition diagram to model a frictionless multibody collision. Each state describes a different topology of the collision characterized by the set of instantaneously active contacts. A change of state happens when a contact disappears at the end of restitution, or when a disappeared contact reappears as the relative motion of two bodies goes from separation into penetration. Within a state, (normal) impulses are coupled differentially subject to relative stiffnesses at the active contact points and the strain energies stored there. Such coupling may cause restart of compression from restitution during a single impact. Impulses grow along a bounded curve with first-order continuity, and converge during the state transitions. To solve a multibody collision problem with friction and tangential compliance, the above impact model is integrated with a compliant impact model (Jia 202). The paper compares model predictions to a physical experiment for the massé shot, which is a difficult trick in billiards, with a good result. KEY WORDs impact, multiple impacts, energy-based restitution, state transition diagram, stiffness ratio, impulse curve, compliance, billiard shooting Introduction Impact is a phenomenon that takes place and is made use of in our daily life. It occurs when two or more bodies collide, establishing a short period of contact. During this period, impact generates an impulsive contact force with magnitude tens or hundreds of times higher than that of a static contact force. We apply impulsive forces to accomplish tasks that would be inefficient or difficult to achieve otherwise. Examples include landing after a jump, putting a cup of coffee on the table, punching a staple through a thick stack of documents, kicking a soccer ball, shooting a billiard, a bullet hitting a target, etc. On other occasions we want to extend impact periods to minimize impulsive forces that could do harm to ourselves. Devices designed for such purpose include air bags in an automobile, baseball and boxing gloves, football helmets, gymnastic mats, etc. Impact has applications in many manufacturing and robotic manipulation tasks that involve high-speed operations. A bowl feeder (Boothroyd and Redford 968) exerts vibrations to channel parts along a spiral inclined track designed to sort them in a small number of fixed orientations. Every part will go through multiple collisions with gates along the track. The earliest work on

2 impulsive manipulation can be traced back to Higuchi (985), who demonstrated linear positioning of tools using electromagnetic impulsive force. Parts with simple geometry floating on an air table can be sorted by hitting them with a rotating stick (Hirai et al. 999). A part can be struck only once (Han and Park 200) or tapped repetitively (Huang and Mason 2000) to reach a desired configuration, by imparting initial velocities calculated through analyses of impacts and the part s post-impact sliding motions. For space robots, collisions are inevitable either with other free-floating objects (Yoshida and Nenchev 995) or from landing on other planets, either inside spacecraft or by themselves. Meanwhile, work has been carried out on making a robot with low degrees of freedom juggle an object via impact. A common objective here is to turn the object s original free flying motion into a vertical periodic one. Rizzi and Koditschek (992) implemented a robotic paddle that was capable of batting a ping-pong ball into a steady periodic motion via controlled impacts. Zavalo-Rio and Brogliato (999) studied the feedback control of a one degree-of-freedom juggling robot by treating it as a mechanical system under some unilateral constraint. Their control design was later extended by Brogliato and Zavalo-Rio (2000) to a class of complementary-slackness juggling systems through integration of such constraint with an impact law. Lynch and Black (200) used the idea of force recurrence to demonstrate that a planar juggler, under controls found via nonlinear optimization, could converge an initial free flight trajectory of a puck to a vertical limit cycle. Despite accomplishments in the design of a practical walking machine (Raibert 986), most existing walking robots are still lab prototypes that perform statically stable gaits at low and constant speeds to avoid inertial effects. Understanding the foot-ground collision can be important for resolving instability due to large inertial forces generated by fast walking and running (Garcia and de Santos 2005). Research efforts on robot dynamics with impact include modeling of collisions between a robot and its environment (Zheng and Hemami 985), design of control schemes for stable contact during such collisions (Volpe and Khosla 993), evaluation of collision effects on a robot (Walker 994), and generation of a large impulsive force by a humanoid robot without losing balance (Konno et al. 20). Impact also has a large potential application in sports robotics. An air hockey-playing robot (Partridge and Spong 999) applies impact to control the trajectory of a puck sliding on an air table with low friction. Skills in ball sports are essentially about understanding how to strike balls to make them follow desired trajectories while moving and spinning fast enough to evade the opponents. A banana kick in soccer, performed by striking the bottom left or right side of the ball, results in a curved trajectory in the air along which the ball could travel past an array of defenders, the goalie, or a post into a net. This is a perfect example of impact with tangential compliance. In pocket billiards (Shamos 993), a shot often results in two concurrent impacts: one between the cue stick and the cue ball and the other between the ball and the table. Tangential compliance between the cue tip and the cue ball is critical for generating desired velocity and spinning of the cue ball with two objectives: to pocket an object ball via collision, and to make the cue ball go to a location where the next shot can continue comfortably. Multiple impacts occur when several bodies collide at more than one contact, or more common in reality, when one body hits another body that is at rest and in contact with several other motionless bodies. Multiple impacts are unavoidable in the simulation of dynamical systems that involve rigid body collisions. They are also common in sports actions such as a billiard shot mentioned earlier (and also a break shot) and a bowling throw to knock down pins. An impulsive force has very short execution time, and therefore good potential for improving 2

3 task efficiency. Its use could considerably simplify the robotic mechanism performing a manipulation task, while avoiding uncertainties accumulated over repeated complex operations otherwise. Despite many potential applications of impact, impulsive manipulation has remained an area in robotics where relatively little work is known, likely because the foundation for modeling rigid body impact is not fully developed and existing theories often seem either too simple to be realistic or too complex to be applicable, especially in the presence of friction and tangential compliance, not to mention nonlinear viscoelastic effects. Discrepancies often exist between an introduced theory and the findings from an experiment intended for its validation. Robotic manipulation is often treated as a multibody system with point contacts. Its state is described by generalized coordinates q that gather the positions and orientations of the rigid bodies (including the links of the manipulator) in the system. It is well known that the system dynamics, combined with the contact kinematics, can be described as M(q) q + C(q, q) + G(q) = F(q, q), () where M is the symmetric and positive-definite mass matrix, C gathers all centrifugal and Coriolis terms, G includes gravity terms, and F represents the generalized force, which includes the applied forces, torques, as well as the effects of friction and compliance. Such effects typically depend on the generalized velocity q. Suppose the rigid bodies in the system are simultaneously engaged in a collision. It is reasonable to consider that the collision has a very small duration τ. We integrate the dynamics equation () over τ. Since the terms C(q, q) and G(q) are bounded, their integrals vanish as τ 0. Also, when τ 0, the configuration q of the system remains unchanged. Hence we have M(q) q = τ 0 F(q, q)dt = A(q)I n (q, q) + B(q)I (q, q), (2) where I n gathers all of the impulses due to the normal contact forces between the rigid bodies, and I gathers all of the impulses due to their tangential contact forces. Typically, the impact configuration q can be computed. This means that M(q) is a constant matrix. Also, since the torques are linear in the contact forces, A and B are constant matrices as well. Solution of the above multibody collision problem requires determination of the post-collision velocities of the rigid bodies from their configuration q and pre-collision velocities q 0. Essentially, this requires computation of the normal and tangential impulse vectors I n and I. We solve the problem via a decomposition as follows: i) At each contact, obtain the tangential impulse from the normal impulse using their differential relationship via an energy-based contact mode analysis. ii) Analyze the differential relationships among the normal impulses at different contacts so we can choose one normal impulse at a time as the sole variable for the collision process. Part i) is a problem of impact with friction and tangential compliance. It is treated in Jia (202) via the introduction of a compliant impact model. Part ii) is a problem of multiple impacts to be investigated in this paper. More specifically, we will study multiple frictionless impacts with the introduction of a state transition model that tracks impulse accumulations at different contacts. Near the end of the paper, we will combine the two impact models to solve the multibody impact besides the aforementioned work, only Izumi and Kitaka 993 and Tagawa et al. 200 to our knowledge 3

4 problem. With this work we hope to enhance the foundation for impulsive manipulation, especially in a situation where multiple impacts happen at the same time between a robot and the objects it is manipulating. Objects engaged in impacts simultaneously may break and re-establish contact multiple times. In our model, a state represents a distinct subset of the contact points that are active within an interval of the entire duration. Figure illustrates a collision that results from a ball with downward velocity striking another ball that is resting on the table. The initial configuration is drawn next to the origin with the centers of the two balls vertically aligned. The collision process is described in terms of accumulations of the impulses I and I 2 at the ball-ball and balltable contacts, respectively. 2 It is decomposed into a sequence of five states S,S 2,S,S 3,S 4, which correspond to five segments of growths of (I,I 2 ): [O,B), [B,D), [D,E), [E,G), and G. In the figure, every state is augmented with a configuration during the state, where active contacts are represented by (virtual) springs. The collision starts out with the upper ball hitting the lower ball and the lower ball in turn hitting the table. In the first state S, both impacts are active. The upper ball s velocity increases (i.e., its downward speed reduces), while the lower ball s decreases. Strain energies stored at the two contacts are increasing until at A, where the ball-ball I 2 O S S S 2 S 3 S 4 E F G D C B 0.4 A Figure : State sequence S, S 2, S, S 3, S 4 and growth of impulses along a curve that results from a collision. Virtual springs represent active contacts. contact begins to restitute, releasing part of its stored energy while losing the other part to irreversible deformation at contact. Restitution ends when the two impulses reach B with the two balls separating, one going upward and the other downward. The second state S 2, with only active ball-table contact, immediately follows. During S 2, the lower ball s velocity reaches zero at point C and continues growing afterward. It eventually catches up with the upper ball s velocity at D, reestablishing the ball-ball contact. Since both contacts are now active, S 2 transitions back to S at D. During S, the ball-table impact finishes at E before the ball-ball impact. The ball-table contact breaks, and state S 3 with only the ball-ball contact active starts. The ball-ball impact ends compression at F. After some loss due to the contact s plasticity, the remaining strain energy is converted into the upper ball s kinetic energy. As restitution ends at G, the terminal state S 4 is reached. Collision ends with both balls going upward and the upper ball at a faster speed.. Paper Outline This paper presents a formal treatment of the model for multiple impacts which was proposed by Jia et al. (2008) with new theoretical developments, and combines the model with a recently 2 Values are properly set to physical parameters including the ball masses, the contact stiffnesses, and the coefficients of restitution of the ball-ball and ball-table impacts. I 4

5 developed model (Jia 200) 3 for three-dimensional impact with friction and compliance. To focus on presenting the model for multiple impacts, we consider the absence of friction and tangential compliance. The dynamics equation (2) reduces to M(q) q = A(q)I n (q, q). (3) As mentioned earlier, both matrices M(q) and A(q) depend on the geometry and configuration of the system, and can be set up for individual problems from dynamics and contact kinematics. We will focus on analyzing the ball-ball-table collision presented in Figure. There are several reasons for such a choice. Turning out to be two-dimensional, the collision is likely the simplest multiple impact problem. Yet, its solution can be easily generalized to solve a three-dimensional problem, like modeling of a massé billiard shot (to be examined later), in which one of the two impacts is eccentric 4 A collision involving three rigid bodies with arbitrary geometry will be handled easily once the dynamic equation (3) is set up. In Section 2, we present a quick review of single impact with frictionless contact, offering the derivative of contact energy with respect to impulse. In Sections 3 to 7, we give a formal treatment of the multiple impact model for the ball-balltable collision. Section 3 presents a transition diagram involving four states, each representing a different combination of active contacts during the collision, and introduces a set of energy-based impact laws. Section 4 describes impact evolutions within the states. Section 5 reveals the effects of the stiffness ratio and the mass ratio, and establishes the scalability of the post-impact ball velocities over the pre-impact velocity of the upper ball. Section 6 introduces the impulse curve as a trajectory of growth in the ball-ball and ball-table impulses, and shows that during the collision the curve lies inside an ellipse derived based on non-negative contact strain energy. Instances of elastic and plastic collisions are illustrated. Section 7 proves the convergence of state transitions. Section 8 integrates the multiple impact model with a model for three-dimensional impact with tangential compliance 5 developed by Jia (202), to simulate the task of shooting a billiard ball. Impact dynamics and contact kinematics are combined with frictional contact modes analyzed according to the second model. Section 9 describes two experiments: one of dropping a ping pong ball onto another and the other of a massé shot. In Section 0, we will give a summary and discuss an extension to simultaneous collisions involving three or more bodies, along with other future research directions..2 Related Work on Multiple Impacts All collisions conserve momentum under Newton s third law. Those that also conserve kinetic energy are elastic, while those that do not are plastic. Molecular collisions can be regarded as elastic. Collisions that we come across in daily life are generally plastic. The problem of determining postcollision velocities is under-constrained by momentum conservation alone. One or more impact laws need to be imposed accordingly. There are three commonly used laws: Newton s law, Poisson s hypothesis, and the energy hypothesis, which respectively specify the ratios between the velocities before and after an impact, the impulse growths during two different impact phases, and the 3 This model is investigated in more details in Jia (202). 4 A central impact between two bodies happens when their centers of mass lie on the common normal line passing through the contact. The impact is eccentric if one of the centers does not lie on the line. 5 Compliance allows part of the work done by the tangential reaction force to be recoverable rather than dissipate entirely due to friction. 5

6 strain energies stored and released during those phases. Jia (202) presents a progressive overview of research in impact mechanics centered around these laws. Here we primarily survey work on multiple impacts, which happen simultaneously but are sometimes not treated exactly so in the analysis. Newton s law of impact (Mason 200, p. 22) asserts that the speed of an object after an impact is a constant fraction of that before the impact. It was applied very early on in the study of multiple impacts by Maclaurin (742), and in particular, in the treatment of elastic collision in a system of spheres by Bernoulli (969). More recently, Ivanov (995) examined four approaches that apply Newton s law to solve multiple impact problems with no friction. He reported that two of the approaches generated unrealistic results or no unique solution, one was highly sensitive to the initial conditions, and one had to consider all possible sequences of pairwise collisions (thus increasing the complexity significantly). Newton s law implies that still objects before an impact should remain still. This is nevertheless incorrect since energy often gets transferred to these objects in the impact duration. Consider a simple version of Newton s cradle (Brogliato 999), where three balls are aligned horizontally with the leftmost ball having an initial velocity to start the collisions. All three balls may get velocities out of the process (Liu et al. 2009). Another counterexample is dropping a ping pong ball onto another one resting on the table. Both balls will bounce up as a result. Newton s law is generally inadequate for modeling multiple impacts and impacts with contact friction and tangential compliance. Poisson s hypothesis (Routh 905) states that an impact between two bodies begins with a compression phase until their approaching velocity decreases to zero, and follows with a restitution phase until the two bodies fully separate. The hypothesis also asserts that the impulse accumulated during restitution is a fraction of that accumulated during compression. Aside from the possibility of predicting an increase in kinetic energy (Wang et al. 992; Stewart and Trinkle 996), Poisson s hypothesis applies to the impact at each contact during simultaneous collisions in an isolated way, whereas the phases of the impacts at different contacts often depend on each other. There may not be enough energy stored at the contact to provide the amount of impulse growth during restitution as prescribed by Poisson s hypothesis (Jia et al. 2008). Impact analysis based on Poisson s hypothesis has been a subject of controversy over its consistency with Coulomb s law of friction and the law of energy conservation. Glocker and Pfeiffer (995) formulated two-dimensional impact under Coulomb friction and Poisson s hypothesis as a linear complementarity problem (LCP). They assumed that impacts at different contacts end compression and restitution simultaneously, and used a polyhedral approximation of the Coulomb friction cone to realize complementarity in the tangential direction (in addition to the non-negativeness of impulse in the normal direction). Impulses at the ends of compression and restitution were obtained under different LCP formulations, by employing Lemke s algorithm (Anitescu et al. 996) which pivoted like the simplex algorithm in linear programming. A similar formulation of multi-rigid-body impact problems with friction was proposed by Stewart and Trinkle (996), who applied a time stepping (integration) scheme that extended to a nonlinear complementarity (NCP) formulation. Their method was modified by Anitescu and Portra (997) in order to guarantee a solution (though multiple ones may exist). We refer to Glocker (200) for a survey of LCP-based methods for frictionless collision problems. An LCP formulation of multiple impacts, however, does not correspond to fundamental physical properties (Chatterjee and Ruina 998). It is based on an unrealistic assumption that all impacts 6

7 synchronize in compression and restitution. Lemke s algorithm only generates the impulse values after compression and restitution, but does not describe how the impulse accumulates, which is important for contact mode analysis during the impact. Also, ambiguities exist because often the LCP solution is not unique. It is worthy to note that none of the frictional impact models mentioned so far under Poisson s hypothesis handles tangential compliance, which would return a portion of the work done by the tangential interaction force during an impact rather than let it dissipate completely under friction. Concurrent multiple collisions are sometimes sequenced into two-body collisions either by order of position (Ivanov 995), by order of normal approach velocity (Chatterjee and Ruina 998), or by a variational approach (Seghete and Murphey 200) that extremizes action over nearby trajectories of successive single impacts. Most of this type of works deal with frictionless impact. High-speed photographs of such collisions show that multiple objects are simultaneously in contact rather than only two at a time (Stewart 2000). Observations also seem to suggest that, during the physical process, the involved objects may have broken and re-established contact multiple times. These coupling effects among impacts at various contacts were approximated by some impulse correlation ratios (Ceanga and Hurmulu 200). Similar studies of a 3-ball chain were conducted by Acary and Brogliato (2003), and by Acary and Taha (2005) with the introduction of a ratio between the total kinetic energies after and before the collision. One problem with the impulse correlation ratio is its lack of precise physical description. Stronge (990) developed an energy-based model that defines the energetic coefficient of restitution as the square root of the portion of the strain energy absorbed during compression that is to be released during restitution. This coefficient measures energy dissipation directly, and is consistent with the law of energy conservation, unlike Newton s impact law and Poisson s hypothesis. Liu et al. (2008) described a framework for frictionless multiple impacts in a multi-body system. Impulses are related to each other differentially, and determined by the relative contact stiffness as well as the ratios of corresponding contact potential energies. Numerical integration is always carried out over the impulse at the contact currently storing the maximum potential energy. Energetic coefficients of restitution are applied to individual impacts, each of which may go through multiple compression-restitution phases. Referred to as the LZB multiple impact model, this work deals with general contact stiffness that subsumes those for Hertz and linear contacts. Their sequel paper (Liu et al. 2009) presented a numerical algorithm and included simulation results for several benchmark problems including Newton s cradle, the Bernoulli problem, etc. As nicely as the theory was presented, it did not analyze the collision behavior in the course of the accumulation of individual impulses, nor did it provide a proof for termination or convergence. Based on the LZB model, Zhao et al. (2008) studied energy dissipation and transfer during multiple frictionless impacts. They obtained numerical results for a column of particles (a granular chain) that were in good agreement with experimental findings (Falcon et al. 998). Nguyen and Brogliato (202) recently used the same model to describe wave propagation during impacts of multiple granular chains, matching some main observations by Nakagawa et al. (2003) from their experiment. Jia et al. (2008) independently proposed a multiple impact model that observes the same differential relationship between impulses at various contacts, keeps track of contact strain energy for restitution, and performs numerical integration over the currently dominant impulse. The main difference from the LZB model is that Jia et al. s model formulates the physical process as a state transition diagram, where each state represents a different combination of contacts that 7

8 are instantaneously active. A transition from one state to another happens when either an active impact finishes restitution or an inactive impact gets reactivated. Every collision instance yields a sequence of states with proven termination/convergence. Jia et al. (2008) also showed that, for a linear arrangement of three objects (the third of which having infinite mass), the impulses grow along a planar curve bounded by an ellipse derived from non-negative strain energy during the collision. As mentioned earlier, this paper extends Jia et al. (2008) in formal analysis, and integrates the model with another model (Jia 202) to handle friction and tangential compliance in multibody collisions in three dimensions. 2 Single Impact with Frictionless Contact This section presents a very brief review of single impact with frictionless contact. Suppose a particle of mass m with downward velocity v 0 < 0 impacts on a horizontal plane. The particle receives a contact force F 0 that varies in the duration τ of the impact. The force is physically continuous, and thus integrable over [0,t], 0 t τ, yielding the impulse I = t 0F dt. Conversely, we have I = F during the impact, where the dot. denotes differentiation with respect to time. Furthermore, a one-to-one correspondence exists between t and I given that F > 0 except at the beginning and end of the physical process. On the particle, the gravitational force is significantly less than the impulsive force F and thus ignored. It follows from Newton s second law that during the impact F = m v, where v is the particle s velocity. Integration of the equivalent equation I = m v establishes that the total impulse is equal to the particle s change in momentum. This yields the velocity equation during the impact: v = v 0 + I m. (4) Under energy conservation, the particle s post-impact velocity cannot exceed v 0. The total impulse is thus finite. As the impact duration τ 0, F in order to keep the integral I finite. It is reasonable to assume that the impact happens in infinitesimal m time. It is thus best analyzed in the impulse space. To better understand v 0 x this physical process, we connect the particle and the plane with a virtual spring with stiffness k. See Figure 2. Let x measure the v change in the virtual spring s length from its rest length. The value of x is negative when the spring is compressed. We have the ball velocity v = ẋ, the contact force F = kx, and the potential energy stored by Figure 2: Particle impacting a table. and finishes. the spring E = 2 kx2. Note that E = 0 only when the impact begins The impact can be divided into two stages (Mason 200, p. 22): compression and restitution. Compression transforms the particle s kinetic energy into the potential energy E of the spring, and ends at ẋ = 0 when the energy reaches its maximum value E max. The impulse exerted up to this point is I = mv 0 by (4). Restitution then releases the elastic portion of the stored energy, of the amount e 2 E max. Here e, 0 e, is referred to as the 8

9 energetic coefficient of restitution, 6 which was analyzed by Stronge (990) 7. The remaining portion ( e 2 )E max is simply dissipated due to some irreversible internal deformations. With the energy release, the impulse grows by an additional amount of emv 0. The impact ends with the particle velocity ev 0. We adopt the explanation by Stronge (2000, p. 98) for the energy loss 8 : When compression ends the spring stiffness suddenly increases from k to k/e 2. Such increase can be attributed to the hardening of the material due to some irreversible deformation. 9 Continuity of F then ensures a simultaneous reduction of the change x in length of the spring to e 2 x. 0 The one-to-one correspondence between I and t establishes E as a function of I, even though the loss of energy after compression is directly due to deformation, not I. Discontinuous only at the end of compression, the function E is differentiable during each impact phase: de di = Ė = d( 2 kx2 )/dt I kx = ẋ (5) = v = ( v 0 + I ), by (4). (6) m The third equation above holds because the stiffness k does not change its value during compression or restitution. Since I is continuous, at the impact phase switch the left and right derivatives of E, both given by (6), are equal. 3 Multiple Impacts From now on until Section 7, we study the following problem. A rigid ball B with mass m and initial velocity v 0 < 0 strikes downward onto another rigid ball B 2 with mass m 2 resting on a horizontal table. The two balls have their centers vertically aligned, and frictionless contacts between them and between B 2 and the table. The scenario is depicted in Figure 3(a). As B collides with B 2, the latter ball in turn impacts the table. The goal is to determine the rebound velocities of the two balls. This seemingly simple problem demands non-trivial analysis (which can be generalized to solve more complex problems). 6 Poisson s hypothesis states that I will accumulate ẽmv 0 more during restitution to yield the final velocity ẽv 0, where ẽ [0, ] is called the kinetic coefficient of friction. When friction exists at the contact and the direction of contact slip varies during collision, the kinetic coefficient of restitution ẽ is not consistent with energy conservation (Stronge 2000, p. 47). The two coefficients of restitution ẽ and e are otherwise equivalent for single impact unless the bodies in impact are rough and the impact configuration is eccentric (Stronge 2000, p. 28). 7 and initially due to Boulanger (939) 8 An alternative is a mono-stiffness model (Goldsmith 960) that does not change the stiffness at the end of compression but rather ends restitution at the energy value E = ( e 2 )E max. This approach, used by both Liu et al. (2008) and Jia et al. (2008), is able to produce cycles of compression and restitution that have been experimentally validated (Antonyuk et al. 200). In our paper, the increase in stiffness after compression reflects the hardening of material better as well as makes it convenient to model transition of compression back to restitution, a phenomenon remarked by Liu et al. (2008) and described later in Section Such material deformation is assumed here to happen much faster than impact. 0 A different bi-stiffness model (Zhao et al. 2008; Nguyen and Brogliato 202) subtracts from the change in length at the end of compression a portion of e 2 due to plastic deformation. 9

10 Let v and v 2 be the respective velocities of B and B 2 during the collision. We attach a virtual spring between B and B 2 with stiffness k, and another one between B 2 and the table with stiffness k 2, as shown in Figure 3(b). The stiffness k i, i =,2, has original value k i. Following the discussion over single impact in Section 2, k i will increase by a factor of /e 2 i whenever compression ends, where e i [0,] is the impact s coefficient of energy restitution. Let x and x 2 be the changes in length of the two virtual springs. The gravitational forces are negligible compared to the large impulsive forces F and F 2 exerted on B and B 2, respectively. The kinematic and dynamic equations are: ẋ = v v 2, (7) ẋ 2 = v 2, (8) m v = F = k x, (9) m 2 v 2 = F 2 F = k x k 2 x 2, (0) v 0 m m 2 (a) B B 2 v v 2 F F 2 m m 2 (b) Figure 3: (a) Ball-ball-table collision and (b) virtual springs for modeling. In the duration τ of the ball-ball-table collision, there are two impulses: I = t 0 F dt and I 2 = t 0 F 2 dt, 0 t τ. Since F,F 2 0, they never decrease. Integrate (9) and (0): v = v 0 + m I, () v 2 = m 2 (I 2 I ). (2) The two virtual springs store strain energies E = 2 k x 2 and E 2 = 2 k 2x 2 2, respectively. Their derivatives assume the form (5), into which we substitute (7), (8), (), and (2): de = ẋ = v 2 v di ( ( = v ) ) I m2 I 2, (3) m m 2 de 2 = ẋ 2 = v 2 di 2 = m 2 (I I 2 ). (4) In the meantime, a differential relationship exists between the two impulses I and I 2 that depends on the two energies 2 : di 2 I = 2 = k 2x 2 k2 E = 2. (5) di I k x k E The ordinary differential equations (ODEs) (3) (5) can be viewed as a system in one variable I (or I 2 ) that governs how the two impacts evolve. We abuse the meaning of derivative at the end of compression to refer to the equal left and right derivatives of the energy E i. The same convention will apply to its second derivative (which is continuous according to Proposition 6). 2 The ratio was called by Liu et al. (2008) as the distributing law and presented for nonlinear contact as well. 0

11 3. State Transition Diagram Though the ball-ball and ball-table impacts begin simultaneously, they will hardly end compression or restitution at the same time. For instance, when the ball-ball impact finishes restitution, the ball-table impact may be still undergoing restitution. As collision continues, the two balls may start moving toward each other again at some later point, reactivating the ball-ball impact. This suggests that we sequence the collision process into (repeats of) three states: S, when both impacts are active; S 2, when only the ball-table impact is active; and S 3, when only the ballball impact is active. The three are referred to as the impact states. We also introduce a fourth state S 4, which is terminal and happens only if all active impacts end restitution simultaneously in S, S 2, or S 3. With two active impacts the system is in S. Both impacts could end simultaneously, placing the system in state S 4. Otherwise, one impact ends before the other, placing the system in either state S 2 or state S 3. In S 2, there is no ball-ball impact, just a ball-table impact. If the ball-table impact causes the lower ball to catch up with the upper ball in velocity, the system then re-enters state S. Similarly, in state S 3, there is only a ball-ball impact. Again, the system re-enters state S if the lower ball starts moving downward. The above idea leads to a state transition diagram shown in Figure 4. The collision always starts with state S. A transition S S 2 happens when the ball-ball impact finishes restitution before the ball-table impact. So the two balls are breaking contact momentarily. Since the ball-ball impact was in restitution just before the transition, ẋ 0, which by (7) implies v v 2 when S 2 begins. Because gravity is neglected during the collision, v will not vary during S 2. The state will transition back to S if v 2 increases to become equal to v before restitution of the ball-table impact ends. Otherwise, state S 4 will be reached with v v 2 to end the collision. Note positiveness of the coefficient of restitution e as part of the condition for the state transition S 2 S. When e = 0, the state S just before S 2 must have ended with compression of the ball-ball impact. There was no restitution phase in S because the strain energy E was lost under e = 0. The two balls collapse into one object, reducing the remaining process to that of a single impact. This case is analyzed in the paper s supplement Jia et al. 20a (Section E.3). Similarly, a transition S S 3 happens when the ball-table impact finishes restitution before the ball-ball impact. At the moment, v 2 0. State S 3 will transition to S when v 2 = 0, that is, when the lower ball re-establishes contact with the table. An extra condition e 2 > 0 is imposed over the transition, since otherwise the lower ball and the table collapse into one body, leading to a single impact problem (treated in Jia et al. 20a). If the ball-table impact finishes restitution in S 3, S 4 follows to end the collision. State S 4 is the terminal state. The only computation within the state happens with plastic ball-ball collision (e = 0) when this impact ends compression in state S 3. We again refer to Jia et al. (20a, Section E.3) for a treatment of this special case. An impact may start with one state, end compression in another, and finish restitution to end a third one. By induction on the number of states, we can easily establish that equations () and (2) hold across any state sequence. Proposition The velocities of the two balls are () and (2) during the collision. The collision described in Figure is generated with m =, m 2 = 2/ 3, k 2 /k =, 3 e = 0.9, 3 Section 5 will show that the collision outcome is affected by the stiffness ratio k 2/k but not individual stiffness values.

12 S v v spring ends v 2 spring 2 ends restitution first restitution first (v v 2 ) 2 (v 2 0) S 2 e > 0 and e 2 > 0 and v = v 2 before v 2 = 0 before S 3 spring 2 ends restitution spring ends restitution v v 2 both springs end restitution together v 2 S 4 (end of collision) otherwise, spring 2 ends restitution (v v 2 0) v otherwise, spring ends restitution (v v 2 0) v 2 Figure 4: State transition diagram for the ball-ball-table collision in Figure 3. During each state S i, i 3, v and v 2 can be either upward or downward, except v 2 is always upward in S 3. Both velocities are upward in S 4. and e 2 = 0.7. The initial velocity of the upper ball is v 0 =. The impulse trajectory takes the shape of a curve 4. The dots A, B, F, G on the curve mark the ends of a ball-ball impact phase, while the dots C and E mark the ends of a ball-table impact phase. The dot D represents a transition S 2 S due to v = v 2 not the end of any impact phase. The coordinates of G are the final impulses exerted on the balls. Substituting them into equations () and (2), we obtain the ball velocities after the collision. In Figure, we see that I does not change in S 2 while I 2 does not change in S Restart of Compression from Restitution From Section 2, single impact starts with compression and ends with restitution. Once started, restitution will never switch back to compression. When two impacts are simultaneously active (i.e., in state S ), however, this is no longer true. The two ball velocities, given in () and (2), are governed by the differential equations (3) (5), the last of which is nonlinear. It may happen that during restitution of the ball-ball impact v v 2 decreases to zero and goes negative. Then 4 which will be formally introduced as the impulse curve in Section 6 2

13 the impact will end restitution and restart compression at v = v 2. 5 Section after the notion of impulse curve is introduced. We defer an example to 3.3 Assumptions of Multiple Impacts We extend the law of single impact in Section 2 to the following set of three assumptions for multiple impacts (for i =,2): i) Whenever the ith impact ends compression, the contact stiffness suddenly increases from k i to k i /e 2 i while the strain energy suddenly drops from E i to e 2 i E i. Or, more succinctly in the assignment forms: k i k i /e 2 i and E i e 2 i E i. ii) The value of k i does not change whenever the ith impact switches from restitution back to compression. iii) Reactivation of the ith impact does not affect the value of k i. Under assumption i), the contact force 2k i E i is continuous when the impact ends compression. So is the impulse derivative di j /di i, j i, which equals the ratio of the two contact forces. When restitution switches back to compression, loading of the ith virtual spring restarts. Under assumption ii), its stiffness does not change as the spring begins absorbing work into internal energy. Neither do the contact force F i and the spring energy E i. Hence the impulse derivative is continuous at this transition. Owing to possible restarts of compression, an impact may go through multiple compression and restitution phases. Suppose the impact has ended compression j times, that is, restitution has switched back to compression j times. Then the stiffness k i = k i /e 2j i under assumptions i) and ii), where k i is the original stiffness value. When an impact gets reactivated, it should respect the effect of irreversible deformation that has happened before. This is reflected in zero change to the contact stiffness under assumption iii). More specifically, at a transition S 2 S or S 3 S, k or k 2 retains its value at the end of the last state S. 3.4 Energy Conservation Unless End of Compression The total energy of the impact system consists of the kinetic energies of the two balls and the strain energies at the ball-ball and ball-table contacts. Energy loss happens only at an end of compression if the corresponding coefficient of restitution is less than one, as stated in the following theorem. Theorem 2 The total energy of the two-ball impact system decreases whenever impact i with e i <, for some i =,2, ends compression, but does not vary otherwise. Proof Under the impact assumptions in Section 3.3 when impact i with e i < ends compression, a loss of the strain energy E i by a factor of e 2 i happens. We now prove that the total energy U = 2 m v m 2v E + E 2 is conserved over a period of collision during which no compression of any impact ends. 5 Such a phenomenon was first observed by Liu et al. (2008). 3

14 From (3) and (4) we obtain: ( d(e + E 2 ) = v 0 di + + ) I di I 2 di + I 2 di 2 I di 2 m m 2 m 2 m 2 m ( 2 = v 0 + I ) di + ) (I di + I 2 di 2 d(i I 2 ). m m 2 = v di + ( ) d m 2 2 (I 2 I ) 2 by () ( ) ( ) = d 2 m v 2 + d 2 m 2v2 2 by () and (2). The above equation implies energy conservation: ( du = d 2 m v 2 + ) 2 m 2v2 2 + E + E 2 = 0. Note that di = 0 in state S 2, and di 2 = 0 in state S Energy-Based Criteria for Impact Phases For convenience, we sometimes refer to the ball-ball and ball-table impacts as the first and second impacts, respectively. Compression of the ith impact ends when ẋ i = 0 and ẍ i > 0. Since ẋ i = de i /di i by (5), ẍ i = (d 2 E i /dii 2) I i, and I i = F i > 0 at the moment, the two sufficient conditions are equivalent to de i /di i = 0 and d 2 E i /dii 2 < 0. Namely, the strain energy stored at the ith contact has reached a (local) maximum. Under (3) and (4), the first order necessary conditions for ends of the ball-ball and ball-table compression are respectively ( v ) I I 2 = 0, (6) m m2 m 2 I I 2 = 0. (7) In state S 2, the ball-ball contact is inactive with I I (0), its value at the start of the state. From (4) we obtain that d 2 E 2 /di 2 = /m 2 < 0. If within the state I 2 reaches I (0) to make (7) hold, compression will end. Similarly, in state S 3, the ball-table contact is inactive with I 2 assuming its value I (0) 2 at the state of the state. From (3) we obtain that d 2 E /di 2 = (/m + /m 2 ) < 0. If I grows to make (6) hold during the state, compression will end. In state S, the end of compression will be conveniently determined using the impulse curve and compression lines introduced in Section Three Impact States To focus our investigation, from now through Section 6 we make several assumptions: a) k 2 /k > 0 and is finite; b) e 0; and c) e 2 0. To save space, the degenerate cases k 2 /k = 0, k 2 /k =, e = 0, and e 2 = 0 are treated in details in the supplement Jia et al. (20a, Section E). 6 We will only make some remarks about these cases when needed. 6 In such a degenerate case, the problem can be simplified with a closed-form solution. For example, if k 2/k = (infinitely stiffer ball-table contact than ball-ball contact), the lower ball can be treated as an integral part of the table so the collision reduces to single impact. 4

15 This section looks at how each of the states S,S 2 and S 3 evolves and how to decide the next state. State S 4 need only handle perfectly plastic ball-ball impact, which is also described in Jia et al. (20a, Section E.3) Before examining the other three states one by one, we look at how to track the strain energies stored at the two contacts within a state. 4. Changes in Strain Energies within a State Let us use the superscript (0) to refer to the value of a physical quantity at the start of a state, and the notation to refer to its amount of change so far in the state. Rewrite the velocities of the two balls () and (2) in terms of the impulse accumulations I = I I (0) and I 2 = I 2 I (0) 2 : v = v (0) + m I, (8) v 2 = v (0) 2 + m 2 ( I 2 I ), (9) With (8) and (9), the energy derivative de /di = ẋ from (3) is rewritten as ( ( de = v (0) v (0) ) ) I m2 I 2. (20) di m m 2 Integration of the above yields the change in E from its value E (0) when the state began: E = ( v (0) 2 v (0) ) I ( + ) I 2 + I I 2 di. 2 m m 2 m 2 I (0) There is one bug with the above formula though. If compression has ended during the growth of I from I (0) to its current value, we must include an energy loss by the amount of ( e 2 )E max, where E max was the value of E at the end of compression. Introduce a two-value variable α such that α = if compression ends at some impulse value in (I (0),I ], and α = 0 otherwise. Then the correct form of the energy change is E = ( v (0) 2 v (0) ) I ( + ) I 2 + I I 2 di + α (e 2 2 m m 2 m 2 I (0) )E max. (2) Similarly, from equations (8) and (9) we rewrite (4) and then integrate it to obtain the energy change from E (0) 2 : de 2 di 2 ( = v (0) 2 + ) ( I 2 I ), (22) m 2 E 2 = v (0) 2 I 2 2m 2 I m 2 I2 I (0) 2 I di 2 + α 2 (e 2 2 )E 2max, (23) where α 2 takes on the value or 0 depending on whether or not the ball-table impact ends compression at some impulse value in (I (0) 2,I 2], and E 2max is the strain energy at the end of compression. 5

16 4.2 State S : Simultaneous Ball-Ball and Ball-Table Impacts Both impulses I and I 2 are increasing in the state according to (5). If S is the start of the collision or follows S 3, I is the primary impulse (variable), and I 2 is the secondary impulse (function of I ). We solve the system (3) (5). If S follows S 2, the two impulses switch their roles. We obtain I as a function of I 2 defined by the equation di k E =. (24) di 2 k 2 E 2 This time we need to solve the system made up of (3), (4), and (24). We first consider the case that S is at the start of the collision, and characterize how I 2, E, and E 2 vary with I. From (3), we have de /di (0) = v 0, and hence the Taylor series E = v 0 I + O(I 2 ). (25) In the above, we used the big-o notation such that O(I 2) < ci2 for some constant c and some small enough value of I. Next, we determine the Taylor series for I 2. Let ρ = (di 2 /di )(δ), where δ is some small enough positive number. The idea is to solve for ρ from equation (5) and then take the limit as δ 0 to obtain the value of di 2 /di (0). With I = δ and I 2 (δ) ρδ, we substitute into (2) and (23) the following impulse integrals: δ 0 I 2 di ρδ 0 I 2 ρ di 2 = 2 ρδ2 and ρδ 0 I di 2 δ 0 I ρdi = 2 ρδ2. (26) The two energies are then approximated as (setting α = α 2 = 0 because no compression ends) E = E v 0 δ ( + ) δ 2 + ρδ 2, 2 m m 2 2m 2 E 2 = E 2 ρδ 2 ( ρ). 2m 2 Plug the above equations into (5) and square both sides. A few more steps of symbolic manipulation lead to a quadratic equation: ρ 2 + b ρ k 2 k = 0, where b = 2 m2v 0 δ m 2 m + k 2 k > 0. Thus di 2 di = ρ ) ( b 2 δ 2 + 4k 2/k b = 2k 2 /k. (27) b + b 2 + 4k 2/k Since 4k 2 /k b as I = δ 0, we have that di 2 di = k 2 2m 2 v 0 k I + o(i ), (28) 6

17 where the small-o notation is used such that lim I 0 o(i )/I = 0. From (28) we have di 2 d 2 I 2 di = 0 and 0 di 2 = k 2. (29) 0 2m 2 v 0 k The ball-table impulse has a Taylor series about I = 0: I 2 = k 2 4m 2 v 0 k I 2 + O(I 3 ). (30) Similarly, from (4) and applying (28) and (30), we obtain for small enough I : de 2 di = de 2 di 2 di 2 di = k 2 2m 2 2 v 0k I 2 + o(i 2 ). This gives us the first three derivatives of E 2 at I = 0, and the Taylor series about the point: E 2 = k 2 6m 2 2 v 0k I 3 + O(I 4 ). (3) Proposition 3 A unique solution to the system (3) (5) exists in state S that starts the collision. Proof Collision starts with I = I 2 = 0 and E = E 2 = 0. The initial changes in I 2, E, and E 2 as I increases are unique according to (30), (25), and (3), respectively. Once I,I 2,E,E 2 > 0, existence and uniqueness of the solution to the system follows from Picard s existence theorem (Wolfram Mathworld 7 ) because the right hand sides of (3) (5) are continuous and have continuous partial derivatives with respect to I,I 2,E, and E 2. The right hand side of equation (5) stays continuous when the ith impact ends compression because the contact forces are continuous. Meanwhile, the discrete loss of E i and increase in k i have no effect over (3) and (4). Similarly, we can derive the Taylor series for the secondary impulse and strain energies in terms of the primary impulse at the start of S when the state succeeds S 2 or S 3. Existence and uniqueness of solution within S can be established. Here, (di /di 2 )(I (0) 2 ) = 0 if S succeeds S 2 and (di 2 /di )(I (0) ) = 0 if it succeeds S 3. With no closed-form solution to (3) (5) in general, the state evolution is simulated via numerical integration 8. At each integration step, we check if compression of either impact has just ended as described in Section 3.5, and if so, scale the corresponding strain energy E i by e 2 i accordingly to recognize the energy loss. The state transitions to S 2 when E decreases to zero during restitution of the ball-ball impact, or to S 3 when E 2 decreases to zero during restitution of the ball-table impact, whichever occurs earlier. In numerical integration, we use (27) instead of (29) so that I 2 increases if S is at the start of the collision. For the other two cases, we initialize the impulse derivatives as follows: di 2k /k 2 di, if the preceding state was S 2 (0) 2, I 2 +δ b 2 + b k /k 2 di 2 2k 2 /k di, if the preceding state was S (0) 3, I +δ b 3 + b k 2/k 7 See 8 For numerical stability, it is better to choose I as the variable whenever k E k 2E 2 and I 2 as the variable otherwise. In Liu et al. (2008), I j, j =, 2, is chosen if E j is the larger one of the two energies. 7

18 where b 2 = 2m 2E (0) 2 δ 2m 2v (0) 2 2 δ + k + m 2 and b 3 = 2m 2E (0) δ 2m 2v (0) 2 δ m 2 m + k 2 k. In the supplement Jia et al. (20a, Sections E. and E.2), S will be augmented to handle the special cases k 2 /k = and k /k 2 = 0. If e i = 0 and impact i ends compression in S, another special treatment is needed. This will also be described in the same supplement (Section E.3). k 2 ( m ) 4.3 State S 2 : Ball-Table Impact Only The state S 2 starts with ẋ > 0, or by (7), v (0) 2 < v (0) since the ball-ball impact has just finished restitution. During the state, v v (0), I 0, E 0, and thus from (9), v 2 = v (0) 2 + m 2 I 2. (32) If v (0) < 0, then v (0) 2 < 0. Thus, state S 2 starts during compression of the ball-table impact. Since compression would not end until v 2 = 0, v 2 will reach v (0) first, triggering a transition to S. If v (0) > 0, there are two cases depending on the phase of the ball-table impact when S 2 starts. Compression. Since ẋ 2 = v 2 by (8), we conclude that restitution will start during the state when v 2 = 0 at I 2 = m 2 v (0) 2 by (32). Substituting this value and α 2 = 0 into (23) to determine E 2, we obtain E 2max = E (0) m 2v (0) 2 2. The next state is S if there is enough strain energy after the loss to increase the velocity of the lower ball B 2 to v (0). This leads to the condition e 2 2 E 2max > 2 m 2v (0) 2. Otherwise, the final state S4 is next to end the collision with B 2 having the velocity v 2 = 2e 2 2 E 2max m 2 = e 2 (0) 2 2E (0) v m 2 Restitution. The ball B 2 will increase its velocity to v (0) to reactivate the ball-ball impact (and switches to the state S ) if 2 m 2v (0) 2 (0) 2 +E 2 > 2 m 2v (0) 2. Otherwise, a transition to state S 4 takes place to end the collision with (0) 2 2E (0) v 2 = v m 2 The above analysis of state S 2 can be augmented with the case where e 2 = 0 and the ball-table impact ends compression in the state. We also need to handle the case k 2 /k = 0 when S 2 is the second state and followed by S 4. These two cases are described in Jia et al. (20a, Sections E.2 and E.3). 4.4 State S 3 : Ball-Ball Impact Only During state S 3, I 2 0, and E 2 0. The lower ball B 2 s velocity has a reduced form from (9): v 2 = v (0) 2 m 2 I, (33) 8

19 while the upper ball B s velocity is still (8). Through an analysis based on the phase of the ball-ball impact during which S 3 starts, the maximum possible accumulation of I within the state can be determined (Jia et al. 20a, Section C) to be ( M I max = ( M ( v (0) 2 v (0) + e v (0) 2 v (0) v (0) 2 v (0) + ) M E(0) ), if S 3 starts during compression, ) ( v (0) 2 v (0) ) M E(0), if S 3 starts during restitution, (34) where M = /( m + m 2 ). A transition S 3 S happens if the lower ball s velocity reduces to zero before S 3 ends, that is, if v (0) 2 I max < 0. (35) m 2 ( In this case, the impulse accumulation within S 2 is I = m 2 v (0) ) 2 0 = m 2 v (0) 2 by (33). If condition (35) does not hold, the state ends with impulse accumulation I = I max. The terminal state S 4 is next. The final ball velocities are obtained by plugging I = I max and I 2 = 0 into (8) and (9). A special treatment is given in Jia et al. (20a, Section E.3) for the case where e = 0 and the ball-ball impact ends compression in S 3. 5 Stiffness and Mass Ratios and Input/Output Scalability Sections 4.3 and 4.4 offered closed-form solutions respectively in states S 2 and S 3 based on the values of the impulses, velocities, and energies entering the states. Section 4.2 established existence of a unique solution in S whether it starts the collision or succeeds S 2 or S 3. Theorem 4 A unique solution exists for the ball-ball-table collision problem based on the state transition diagram in Figure 4. Having described the states, we now take a high level view of the ball-ball-table collision as a system with input being the initial velocity v 0 of the upper ball B, and output (or outcome) being the final (separation) velocities v and v 2 of the two balls. The theorem below states how the collision outcome is influenced by the physical parameters. Theorem 5 The following statements hold about the collision outcome: i) It depends on the stiffness ratio k 2 /k but not on individual values of k and k 2. Scaling of k and k 2 by the same factor does not change evolutions of the two impulses I and I 2. ii) It depends on the mass ratio m 2 /m but not on individual values of m and m 2. Scaling of m and m 2 by a factor of s results in scaling of I and I 2, and the strain energies E and E 2, all by s, during the collision. iii) The output-input velocity ratios v /v 0 and v 2 /v 0 are invariant to any change in v 0. Scaling of v 0 by a factor of s will result in scaling of I and I 2 by s, and E and E 2 by s 2 during the collision. 9

20 In fact, part i) of Theorem 5 is a known result. Brogliato (999, pp ) applied a time-based analysis to establish that the separation velocities of three balls colliding in a chain configuration depend on the relative stiffness at their two contact points. Part ii) of the theorem can be interpreted differently. Suppose we impose two unilateral constraints: a) the distance between the centers of the two balls is no less than the sum of their radii; and b) the distance between the center of the lower ball and the table is no less than its radius. The kinetic angle, defined in Brogliato (999) as that between the normals of the corresponding equality constraints, depends on the mass ratio but not on individual masses. The geometry of the collision is determined by the mass ratio (up to scaling), as we will soon examine in Section 6. Following the theorem, we see that the ratios v /v 0 and v 2 /v 0 depend on the stiffness ratio k 2 /k, the mass ratio m 2 /m, and the coefficients of restitution e and e 2 only. Input/output scalability over the initial upper ball velocity and dependence on the mass ratio are two desired properties for an impact model using linear stiffness. They also simplify our analysis of the collision process and computation of the final ball velocities. To solve a collision instance with initial velocity v 0 < 0, and ball masses m and m 2, we instead consider another instance where the upper ball has unit mass and unit initial velocity, and the lower ball has mass m 2 /m. Suppose the new instance yields final ball velocities v and v 2. Then the original collision instance yields final velocities v 0 v and v 0 v 2. The state sequences from the two instances are exactly the same under a one-to-one correspondence between their impulse growths. 6 The Impulse Curve From now on, we need only consider a collision instance where m = and v 0 =. (36) Under Theorem 5, analysis of impulse evolution will carry over for arbitrary v 0 < 0 and m > 0 merely by scaling the impulses and velocities by a factor of m v 0 and the strain energies by a factor of m v 2 0. Accumulations of the ball-ball impulse I and the ball-table impulse I 2 are best described in the impulse plane with I and I 2 as coordinates. Every point in this I -I 2 plane represents a pair of impulse values that may or may not be attained during the collision. By Proposition, the point would yield intermediate ball velocities according to () and (2). As I and I 2 accumulate, the moving point (I,I 2 ) traces out a set of points in the plane. Since the contact forces F and F 2 are continuous and non-negative, the impulse I and I 2 as their integrals are continuous and monotonically increasing. The impulse curve consists of all pairs of intermediate impulse values (I,I 2 ). It grows upward and/or rightward. The next proposition establishes the first order continuity of the impulse curve. Proposition 6 The impulse curve is first order continuous, namely, its tangent is defined everywhere as the curve extends with the growing impulses during the collision. Proof The impulse derivative di j /di i = F j /F i, where I i and I j are the primary and secondary impulses, respectively, is continuous within state S. Within state S 2 or S 3, the secondary impulse I i is constant, and the derivative di i /di j = 0, j i, is clearly continuous. 20

21 Now we show that the tangent is defined at every possible state transition by establishing the equality between the left and right derivatives of the secondary impulse with respect to the primary impulse. There are four types of transitions among states S, S 2, and S 3 as shown in Figure 4. S S 2. At the end of S, the two balls break contact with E = 0. Thus the left derivative di /di 2 = k /k 2 E /E 2 = 0. Throughout state S 2, I remains constant, which implies the right derivative di + /di 2 = 0 at the transition point. S S 3. Similar to the above case. S 2 S. The left derivative di /di 2 = 0 since I does not change its zero value within S 2. The right derivative di + /di 2 = k E k 2 E 2 = 0 because E = 0 at the start of S. S 3 S. Similar to the above case. A transition to state S 4 ends the impact right away Bounding Ellipse In this section, we will show that the two growing impulses are bounded inside an ellipse determined by m 2. To be derived from non-negativeness of the total contact strain energy, this bounding ellipse will help us not only visualize impulse growth during each state but also prove convergence of state transitions later in Section 7. The next lemma bounds the total strain energy using the impulses. Theorem 7 The following inequality is satisfied during a collision where the upper ball has unit mass (m = ) and unit initial velocity (v 0 = ): E + E 2 I 2 I2 2m 2 (I I 2 ) 2. (37) Proof By induction. The energy equations (2) and (23), after substitution of (36), become inequalities with removal of the terms containing α and α 2 from their right hand sides: E ( v (0) 2 v (0) ) I ( + ) I 2 2 m + I 2 m 2 I (0) I 2 di, (38) E 2 v (0) 2 I 2 I2 2 + I2 I di 2. (39) 2m 2 m 2 I (0) 2 From d(i I 2 ) = I 2 di + I di 2 we have I I (0) I2 I 2 di + I di 2 = I I 2. I (0) 2 9 We are assuming e, e 2 0 through Section 6. Otherwise, state S 4 needs to handle the special case where the ball-ball impact is plastic (e = 0) and ends in S 3 (cf. Jia et al. 20a, Section E.3). The impulse curve has a continuous tangent during S 4. 2

22 Add up the inequalities (38) and (39): E + E 2 = ( v (0) 2 v (0) ( ( I (0) 2 I (0) m2 ) I v (0) 2 I 2 ) + I (0) 2m 2 ( I I 2 ) 2 2 I2 ) I ( I (0) 2 I (0) ) I 2 m 2 2 I2 ( I I 2 ) 2, (40) 2m 2 where in the last step we substituted v (0) = + I (0) and v (0) 2 = (I (0) 2 I (0) )/m 2 by () and (2). In the initial state S, v (0) =, v (0) 2 = 0, I (0) i = 0, I i = I i, and E i = E i, for i =,2. This simplifies (40) to E + E 2 I 2 I2 2m 2 (I I 2 ) 2. Next, assume that (37) holds after the first k states; namely, at the start of the (k +)-st state, E (0) + E (0) 2 I (0) 2 ( 2 I(0) I (0) I (0) ) m 2 Add the above inequality to inequality (40), merging the terms on the right hand side: E + E 2 = E (0) + E (0) 2 + E + E 2 (I (0) + I ) 2 (I(0) + I ) 2 ( (I (0) + I ) (I (0) 2 + I 2 ) 2m 2 = I 2 I2 2m 2 (I I 2 ) 2. ) 2 From the proof we see that (37) becomes an equality if and only if there has been no energy loss. This is the case throughout the collision if e = e 2 =. Otherwise, no energy loss occurs until the first impact, with coefficient of restitution less than one, ends compression. Since E +E 2 0, condition (37) imposes a constraint over the ball-ball and ball-table impulses. Corollary 8 Every point on the impulse curve satisfies Appendix A shows that the equation 2 I2 + 2m 2 (I I 2 ) 2 I 0. (4) 2 I2 + (I I 2 ) 2 I = 0 (42) 2m 2 defines an ellipse (see Figure 5) centered at (, ). The semi-minor axis rotates from the x-axis through an angle θ = 2 arctan ( 2 m 2 ). (43) 22

23 I m 2 +m2 2 + m 2 + θ 2 +m2 Figure 5: The impulse curve must lie inside an ellipse and above the I -axis. Here m = and v 0 =. I The ellipse is tangent to two vertical lines: the I 2 -axis at the origin and I = 2 at the point (2,2). It is also tangent to two horizontal lines: I 2 = + m 2, and I 2 = + + m 2. During the collision, the growing impulses (I,I 2 ) must stay inside the region bounded by the ellipse (42) and above the I -axis. This region, shaded in Figure 5, is referred to as the impulse region. Rewrite the velocity equations () and (2) into the matrix form (note m = and v 0 = ): ( ) ( ) ( ) 0 I v + =. m 2 m 2 I 2 v 2 Since the coefficient matrix is non-singular, different points in the impulse region, if representing the impulses at the end of the collision, would generate different final ball velocities. Given the same lower ball mass, every triple (k 2 /k,e,e 2 ) defines a different collision instance. Thus, the impulse region bounds a three-dimensional set of impulse curves resulting from all possible collisions. Every point in the region potentially lies on the impulse curve of some collision instance. It is easy to see that the result on bounded impulse curve generalizes to a collision at n > 2 contacts by deriving an inequality for the total contact strain energy in terms of impulses. 6.2 Partitioning the Impulse Region In the impulse plane, we add two lines l and l 2 defined by equations (6) and (7), respectively. Under (3) and (4), these two lines have equations v 2 v = 0 and v 2 = 0. They respectively represent the first order necessary conditions for ends of compression of the ball-ball and ball-table impacts. Referred to as the compression lines by convention (Mason 200, p. 25), they partition the impulse region into four smaller regions I IV as shown in Figure 6(a). Inside each region is a different combination of the signs of v v 2 and v 2, or equivalently, according to (7) and (8), of the signs of the change rates ẋ and ẋ 2 of the virtual contact spring lengths, and thus a different combination of the phases of the two impacts. The interior of region I includes all impulse values that could possibly be attained during compression of both the ball-ball and ball-table impacts. Accordingly, the region is labeled CC2 in Figure 6(b). The interior of region II includes all impulse values that could possibly be attained during restitution of the ball-ball impact and compression of the ball-table impact. The region is thus labeled RC2. And so on. The impulse curve evolves from the origin at which it is tangent to the I -axis since di 2 /di has initial value 0 by (29). The curve enters region I as both impulses increase. The curve later crosses l (l 2, respectively) where de /di = 0 (de 2 /di = (de 2 /di 2 ) (di 2 /di ) = 0, respectively). It is not difficult to picture that in state S, both impacts could be in either compression or restitution. In state S 2, v > v 2 holds because the two balls are instantaneously separate. So the curve grows vertically upward to the right of l. In S 3, the lower ball has velocity v 2 > 0 since it is separate from the table. The curve grows horizontally rightward to the left of l 2. 23

24 l : ( ) + + m I I 2 m 2 = 0 2 ẋ < 0 ẋ > 0 I 2 III v < v 2 v 2 > 0 IV v > v 2 v 2 > 0 l 2 : I = I 2 I 2 C R2 R R2 ẋ 2 > 0 ẋ 2 < 0 I v < v 2 v 2 < 0 m 2 +m 2 v > v 2 v 2 < 0 II I C2 C R C2 I l 2 m 2 l (a) (b) Figure 6: Regions of four different combinations of impact phases. When a state finishes with the value (I,I 2 ) inside region IV, v > v 2 > 0, state S 4 follows to end the collision. During the collision, E + E 2 > 0. Thus, by (37) the inequality (4) is strict. It is also strict after the collision if there has been some energy loss. Therefore, after the start, the impulse curve will not reach the bounding ellipse (42) again unless the collision ends and e = e 2 =. 6.3 Collision Examples This section presents several collision instances: an elastic one, a plastic one, and a third one in which compression restarts during restitution. In all instances, the upper ball has unit mass (m = ) and unit downward initial velocity (v 0 = ) Elastic Collision Figure 7 illustrates the growth of the impulse curve during a collision with additional physical parameter values specified as follows: m 2 = 2 3, k 2 k =, and e = e 2 =. (44) The bounding ellipse has its minor axis rotated clockwise from the x-axis through π/6. The equations of the ellipse and the two compression lines l and l 2 are marked in the figure. The dots 24

25 A,D,H,J on the impulse curve each marks the end of a ball-ball impact phase, while the dots E and G each marks the end of a ball-table impact phase. The dot F represents a transition from state S 2 to state S that is triggered by a velocity condition (v = v 2 ) not by the end of an impact phase. Starting at the origin tangent to the I -axis and entering region I, the impulse curve crosses the compression line l at the point A. By (3), the two balls have the same velocity, i.e., v = v 2, at the point. Since the curve is entering region II, the ballball impact has finished compression. The curve continues growing to reach the point B, at which its tangent is parallel to the compression line l 2 with slope. At B we infer di 2 /di = from the equation of l 2, and dv 2 /di = 0 from (2). Since the curve slope is increasing at B, so is dv 2 /di. Hence d 2 v 2 /di 2 > 0 holds at B, which implies that v 2 has reached a local minimum value. The value turns out to be an absolute minimum as the only local minimum below l 2 (where v 2 < 0). In other words, the lower ball has the largest downward velocity when the two impulses are at B. The next point of interest on the curve is C, at which the tangent is parallel to l with slope + 2/ 3. Differentiating the last equation in (3), we obtain d di (v 2 v ) = m 2 ( ) di2. (45) di O I III A I B S S 2 S ( l : F C E D G H S 3 IV II ) I 3 2 I2 = 0 S 4 J l 2 : I I 2 = (I I2)2 + 2 I2 I = 0 Figure 7: Elastic collision. The values of the physical constants are given in (44). There are five states sequentially: S over the impulse segment [O, D), S 2 over [D, F), S over [F, G), S 3 over [G, J), and S 4 at J. Since di 2 /di = (+ 3/2)/( 3/2) at C, the same as the slope of l, the derivative of v 2 v vanishes. At the point, the curve slope is increasing, so d 2 I 2 /di 2 > 0. This suggests d2 (v 2 v )/di 2 > 0 according to (45). Thus, v 2 v has a local minimum at C. Since v > v 2 to the right of l 2, the lower ball s velocity is the most below the upper ball s. The ball-ball impact ends restitution when the impulse curve reaches D. State S 2 follows immediately. During the state, the ball-table impact ends compression when it crosses the line l 2 at E. As I 2 continues to accumulate, the curve extends to F, where a transition S 2 S happens. The ball-table impact ends restitution at G with a transition S S 3. In state S 3, the ball-ball impact as the only active one ends compression when the impulse curve crosses l at H and ends restitution at J = (.94484, ), which lies on the bounding ellipse since there is no energy loss under e = e 2 =. The velocities after the collision are v = and v 2 = , respectively Plastic Collision We change the coefficients of restitution to e = 0.9 and e 2 = 0.7 while keeping the values of the other collision parameters in (44). Now, neither impact is elastic. This is the collision instance I 25

26 E 2 v C 2 B O E 3 D C 4 F G B A 0.6 v O C D 3 F E F 4 A A E (a) velocity curve (b) energy curve Figure 8: Collision with energy loss. Diagrams (a) and (b) show the evolutions of the ball velocities and the strain energies at the two contacts. Here m =, m 2 = 2/ 3, k 2 /k =, e = 0.9, e 2 = 0.7, and v 0 =. state interval I I 2 v v 2 E E 2 total energy phases S [O,B) C-R S 2 [B,C) C2 S [C,E) R2 S 3 [E,G) C-R S 4 G Table : Values of I, I 2, v, v 2, E and E 2 at the beginning of every state during the collision depicted in Figures and 8. shown in Figure. With the impulse growth already overviewed following that figure, we focus on how the ball velocities and the strain energies evolve as the impulses accumulate. The collision goes through a sequence of five states: S,S 2,S,S 3, and S 4. They correspond to impulse curve segments [O,B), [B,D), [D,E), [E,G), and G in Figure. Owing to the energy loss, the ending point G is now inside a bounding ellipse, the same as the one in Figure 7. Part (a) of Figure 8 plots the varying ball velocities against each other, while part (b) plots the varying strain energies stored at the two contacts. The velocity curve in (a) is continuous, following the continuity of the impulse curve and the velocity formulae () and (2). Points on the three curves in Figure, and Figures 8(b) and (c) that are labeled by the same letter correspond to the same instant during the collision. On the latter two curves, i labels all segments of impulse evolution within the ith state in the sequence. The velocity curve starts at (,0) in (a), and the energy curve at the origin in (b). The energy loss (of E ) by a factor e 2 = 0.9 happens at the point A when the ball-ball impact finishes compression. It is shown as a discontinuity of the energy curve at A in (b). After the loss, the 26

27 curve resumes at the point A to the left of A. In Table, each row lists the values of the impulses, the ball velocities, and the strain energies at the start of a state in the sequence. For example, the second to last entry in the row of S 2 gives the total energy 2 m v m 2v E + E 2 = of the system at the end of the first state S, following the energy loss at A. The last entry in the previous row, C-R, marks that impact (i.e., the ball-ball impact) has finished compression and restitution within the state S. In the second state (S 2 ) starting at B with only the ball-table impact active, the velocity and energy curves both evolve vertically from B upward since I, v, and E do not change their values. The energy curve, however, has a discontinuity at the point C when the ball-table impact ends compression. Here E 2 suffers a loss by the factor of e 2 2 = 0.5. The curve resumes at C below on the E 2 -axis. When the velocity curve reaches D at which v = v 2, the ball-ball impact gets reactivated immediately to end S 2. Note the small decrease of E 2 over [C,D]. The third state S finishes with restitution of the ball-table impact and no energy loss. 20 In the following state S 3, the velocity curve extends with slope /m 2 according to (33). The energy curve segment, labeled 4 in (b), is on the E -axis. It first grows to the right to reach F where compression of the ball-ball impact ends. Restitution starts at F following an energy loss, and releases the remaining strain energy. The energy curve comes back to the origin with the final ball velocities and at G in (a) and in the last row of Table Restart of Compression Figure 9 shows the impulse curve of a collision with only three states: S, S 3, and S 4. In S, the impulse curve crosses the compression line l three times, at A, B, and D, respectively. The first and third crossings are from left to right, indicating ends of compression. Each of them results in increase of the contact stiffness k by a factor of /e 2. The second crossing indicates a switch from restitution back to compression with no change to the value of k. Note that in S the impulse curve passes through all four regions. The impulse curve also crosses the compression line l 2 at C and E within state S. The balltable contact becomes stiffer by /e 2 2 after C. Similarly, the second crossing point E corresponds to a switch from restitution to compression of the ball-table impact. 7 Convergence of State Transitions Every state terminates. This is trivial for states S 2 and S 3 in which the impulse accumulations I 2 and I are given in closed forms, respectively, in Sections 4.3 and 4.4. In state S, the primary impulse must stop increasing because the growth of (I,I 2 ) is bounded by the ellipse (42). Special collisions (with k 2 /k = 0, k 2 /k =, e = 0, or e 2 = 0) all terminate after a few state transitions (Jia et al. 20a, Section E). In this section we look at the general case where k 2 /k is finite and non-zero, and e and e 2 are non-zero. Under Theorem 5, we need only consider a collision instance with v 0 = and m =. Let s,s 2,... be the sequence of states that results from the collision. Denote by I i and I 2i the respective values of the two impulses I and I 2 at the end of the ith state s i. The sequence {(I i,i 2i )} is monotonically non-decreasing. In the case that it is finite, from the diagram in 20 Due to a round-off error in numerical integration, a slight discrepancy of can be seen in the total energy column of Table between the second S row and the following S 3 row. 27

28 I 2 l 2.4 l S 4.6 D E S 3.2 S 0.8 C B 0.4 A I 0.4 Figure 9: Restarts of compression during restitutions of the ball-ball impact at B and the ball-table impact at D. The lower ball has mass m 2 = 0.5, while the other collision parameters have the same values as for the instance in Figure 8. Figure 4, state transitions will terminate with v v 2 0. In other words, the impulse curve will stop its growth in region IV shown in Figure 6(a), with its boundary included. Suppose that the monotonically increasing sequence is infinite. Bounded inside the ellipse (42), it must converge to some point (I,I 2 ) by a known result from calculus (Kaplan 99, p. 73). Lemma 9 The sequence {(I i,i 2i )} of pairs of impulses ending the states s,s 2,... will not converge to a point in the interior of region I, II, III, or IV drawn in Figure 6(a). Proof Suppose the sequence is infinite and converges to some point q in the interior of one of the regions, say, IV. There exists some integer m such that, for all i > m, the points (I i,i 2i ) are within some neighborhood N of q that is inside the region. However, because the sequence is infinite, state S must transition to S 2 or S 3. State S 2 and S 3 each has single impact, and the impulse curve will always reach the compression line l or l 2 at the end of such a state when it transitions to S. The sequence must therefore exit the neighborhood N at the end of every other state at least. A contradiction. Following Lemma 9, an infinite sequence {(I i,i 2i )} must converge to a point (I,I 2 ) on the compression line l or l 2. It only makes physical sense to have an outcome with both balls rebounding and the upper ball at a speed no less than that of the lower ball. The convergence point (I,I 2 ) should be at or above (,) to yield v v 2 0. To show that (I,I 2 ) is above (,), we will use the following result regarding the minimum increase of the ball-ball impulse or the ball-table impulse within state S during compression. Lemma 0 State S will not end 28

29 I 2 l : v = v 2 IV l2 : v 2 = 0 N(ǫ) l 2 III p = (, ) III N(ǫ) q II q = (I, I 2 ) s j s j+ I v 2 > 0 I v 2 < 0 q j I (a) (b) Figure 0: Contradiction from the hypothesis that the state sequence converges to a point q on l 2 below the intersection p of the compression lines l and l 2 : (a) The sequence is supposed to alternate between S and S 3 inside a disk N(ǫ); meanwhile, (b) the impulse curve must exit the disk (enlarged for display). i) if the accumulation of the ball-ball impulse within the state satisfies I v (0) + m 2 ( + k 2 k ) (46) in the case that S is the first state (with v (0) = ) or follows state S 3 with v (0) < 0, or ii) if the accumulation of the ball-table impulse within the state satisfies I 2 m 2 v (0) 2 + k k 2 ( + m 2 ) (47) in the case that S follows state S 2 with v (0) 2 < 0. Note that v (0) 2 = 0 at the transition S 3 S and v (0) = v (0) 2 at S 2 S. The proof of Lemma 0 is rather involved, and postponed to Appendix B, without interrupting our flow of presentation. Proposition If the sequence {(I i,i 2i )} is infinite, it must converge to a point on l or l 2 but not below (,). Proof By contradiction. Lemma 9 states that the point of convergence q = (I,I 2 ) must be on either l or l 2. Suppose that the point q lies below p = (,). Representing a pair of final impulse values, it would yield the final ball velocities v and v 2. First, we suppose that q is on l 2 below p. Then q lies to the left of l, as shown in Figure 0(a). We have v < v 2 = 0. Denote by N(ǫ) an open disk centered at q and with radius ǫ. This radius can be chosen small enough such that the disk lies completely to the left of l. Every impulse point in N would yield v < v 2 (cf. Figure 6(a)), and v < 0 from () since I <, m =, and v 0 =. It follows from the convergence of the sequence that there exists some integer σ > 0 such that (I i,i 2i ) lies in N for i > σ. Because v < v 2 for i > σ, the ball-ball contact remains in compression 29

30 throughout the states s σ+,s σ+2,... So we infer that s i S 2 (in which the ball-ball impact is inactive), for i > σ. From then on the state sequence must be alternating between S and S 3. Let us consider some state s j+ = S, where j > σ. It begins with the impulse curve at q j = (I j,i 2j ) inside N(ǫ), as shown in Figure 0(b). The point q j is on l 2 because v 2 = 0 at the transition( S 3 S. By ) part i) of Lemma 0 the impulse I will increase by an amount of I v j / + m 2 ( + k 2 k ) in the state s j+, where v j is the value of v as the state s j ends at q j. We can choose ǫ small enough such that I > 2ǫ holds (this is described in the next paragraph). Consequently, I j + I is greater than I + ǫ, which bounds from above the I -coordinates of all points inside the disk N(ǫ). In other words, the impulse curve must exit the neighborhood within the state s j+. Hence a contradiction. Now we look at how to choose ǫ. The condition I > 2ǫ is satisfied if v j + m 2 ( + k 2 k ) > 2ǫ, i.e., v j < 2ǫ ( + m2 ( + k )) 2. k Since q j N(ǫ), I j < I + ǫ, which is equivalent to ( v j < v + ǫ )) by () with m = and v 0 =. It suffices to make sure that v + ǫ < 2ǫ + m 2 ( + k 2 k, which is satisfied when ( ǫ < v / m 2 ( + k 2 k )). Next, we show that the convergence point q cannot lie on l below p again, by contradiction. Suppose otherwise. So q lies on l below p. The states will eventually be alternating between S and S 2 with the impulses in some neighborhood N(ǫ) of q that is below l 2. Inside N(ǫ), at a transition S 2 S, v = v 2 = u for some u < 0 because the impulse point is on l but below l 2. The ball-table impact is in compression at the moment since u < 0. The ball-ball impact immediately goes into compression as v 2 increases to make v v 2 < 0. Hence, state S starts with both impacts in compression, and the impulse curve leaving the line l and going into region I. By part ii) ( of Lemma 0, within ) state S the impulse I 2 will grow by an amount of at least I 2 = m 2 u/ + k k 2 ( + m 2 ). Again, we just need to choose ǫ small enough such that u is close enough to v to satisfy I 2 > 2ǫ. Within the state the impulses (I,I 2 ) will then exit the neighborhood N(q). A contradiction to the convergence at q. The point (, ) represents the outcome of plastic ball-ball and ball-table impacts where both balls have zero velocities after collision. Proposition therefore implies that an infinite sequence of states results in a collision outcome equivalent to that when one of the two impacts is plastic. Finally, we combine Proposition for a general collision with the results on the output velocities in special collisions treated in Jia et al. (20a) Theorem 2 The state transitions will either terminate with v v 2 0 or the impulse sequence will converge to generate either v = v 2 0 or v > v 2 = 0. 8 Shooting a Billiard Integration of the Models for Multiple Compliant Impacts The model described in Sections 3 7 characterizes impulses along the normals at two contacts where impacts take place simultaneously. In the presence of friction and tangential compliance, at each contact, a tangential impulse also exists and has a differential relationship to the normal 30

31 impulse. Only part of the work done by the tangential force dissipates due to friction. The remaining part gets converted into strain energy to be released later. A model for 3-dimensional impact with tangential compliance has been developed by Jia (200, 202) to compute tangential impulse from normal impulse. Integration of the two models will equip us to solve a general impact problem involving multiple frictional and compliant contacts. In this section, we describe how such integration can be done via modeling a shot in the game of pool. As shown in Figure, during a shot by the cue stick at the cue ball, the cue-ball impact and the ball-table impact happen simultaneously at the cue-ball contact point p and the ball-table contact point q. Denote by ˆn the unit normal of the ball at p, and ẑ that of the table at q. At p two unit tangent vectors û and ŵ form a right-handed contact frame with ˆn such that û ŵ = ˆn. Similarly, at q the unit tangent vectors ˆx and ŷ form a right-handed contact frame with ẑ. Let the cue stick have mass m c, and the cue ball have mass m and radius r, thus moment of inertia 2 5 mr2. The cue stick shoots at a velocity v c0 in the unit direction ĉ = v c0 / v c0 that is aligned with the cue s axis of symmetry. The cue-ball impact is eccentric when ˆn and ĉ are not collinear. The condition ˆn ĉ < 0 must hold for the shot to happen. To ensure the ball-table impact to be active during the shot, we first assume that the point p of hit is above the ball s equator (i.e., ˆn ẑ > 0) or on the equator with ĉ ẑ < 0. We also assume that during the shot, the cue stick is constrained to move along ĉ or ĉ. This way our analysis is not overcomplicated by taking into account rotation or bending of the cue stick or shearing of the cue tip during cue ball ẑ q p ˆn û ŷ I 2 cue stick ŵ ˆx I v c0 Figure : Pool shot results in simultaneous cue-ball and ball-table impacts at p and q. the shot, although these effects occur during a real pool shot. This second assumption is satisfied by our design of a mechanical cue stick to be described in Section 9.2. During the shot, the cue stick has velocity v c, the ball has velocity v and angular velocity ω, all varying as the cue-ball and ball-table impacts develop. Denote by I and I 2 the impulses exerted on the cue and the ball at p and q, respectively. The velocity equations are as follows: v c = v c0 + m c (I ĉ)ĉ, (48) v = m (I 2 I ), (49) ω = 5 2mr (ˆn I + ẑ I 2 ). (50) Let v be the relative contact velocity of the cue stick to the ball, and v 2 be that of the ball to the pool table. It follows from contact kinematics with substitutions of (48) (50) that v v 2 = v c v ω (rˆn) ( = v c0 + ĉĉ T + 7 mc 2m 5 2m ˆnˆnT = v ω (rẑ). (( (ˆn ẑ) 5 ) 2 ˆnẑT I + = m ) I + ( + 5 m 2 (ˆn ẑ) 5 ) 2ẑˆnT I 2, (5) ( ẑẑT ) I 2 ). (52) 3

32 ˆn cue tip tangent plane p ŵ ball ball û ẑ ŷ q ˆx table plane (a) (b) Figure 2: Contact structures attaching massless points p and q via virtual springs to (a) the cue tip and (b) the ball, respectively. In the above equations, scalars appearing in the sums with matrices are treated as multiples of the 3 3 identity matrix. Such a sum is always multiplied with a vector so it can be viewed as multiplications distributed into the sum. In (48), an impulse due to the implicit constraint force, which enforces the linear motion constraint on the cue stick, cancels out the component of I orthogonal to ĉ. Decompose the two impulses in their respective contact frames: I = I u û + I w ŵ + I nˆn and I 2 = I xˆx + I y ŷ + I z ẑ. (53) We have I n > 0 or I z > 0 whenever the corresponding impact has been active, and I n = 0 or I z = 0 otherwise. 8. Embedding the Compliant Impact Model The tangential impulses I u û+i w ŵ and I xˆx+i y ŷ are due to compliance and friction and dependent on the normal impulses I n and I z, respectively. To model them, we use the 3-dimensional contact structure from Jia (202), which extends the planar structure introduced by Stronge (2000, p ). As illustrated in Figure 2(a), we view the cue-ball contact p as a massless particle connected to the cue tip via three virtual springs respectively aligned with û, ŵ, and ˆn. For convenience, they are referred to as the u, w, and n springs, while u,v, and n also denote the changes of their lengths by a slight abuse of notation. The infinitesimal area on the cue tip attached to the springs may be thought of as concave. Similarly, part (b) of the figure shows q as a massless particle connected to the cue ball via three springs in the directions of ˆx, ŷ, and ẑ. These springs are referred to as the x, y, and z springs with changes of length x, y, and z, respectively. The impulses I and I 2 result from compressions and extensions of these six springs over the shot period. Let k n and k z be the stiffnesses of the n and z springs with original values k n and k z, respectively. The u and w springs at p have the same stiffness whose ratio to k n is determined by the material 32

33 di n di z or di z di n I n Multiple Impact Model v ˆn I Impact Dynamics I u, I w v c, v, ω di u di n, di w di n Compliant Impact Model (Cue-Ball) v Contact Kinematics I z v 2 ẑ I 2 I x, I y di x di z, di y di z v 2 Compliant Impact Model (Ball-Table) Figure 3: Integrating the models for multiple and compliant impacts. properties of the cue tip and the ball. The x- and y-springs at q also have the same stiffness with a ratio to k z determined by the material properties of the ball and the table. The two normal springs vary their lengths at the rates ṅ = ˆn v and ż = ẑ v 2. (54) The cue-ball and the ball-table impacts are each described by a copy of the compliant impact model (see Jia 202). Each copy is completely characterized by the coefficient of friction and the ratio of the normal stiffness to the tangential stiffness at the same contact. The sliding velocities of p and q in their respective contact tangent planes are (cf. Figure 2) v ṅˆn uû ẇŵ and v 2 żẑ ẋˆx ẏŷ. If one of the sliding velocities is zero, the corresponding contact sticks; otherwise, the contact slips. Mode analysis can be performed for each contact based on the energies stored by its normal and tangential springs. The rates u,ẇ,ẋ, and ẏ of changes in length of the tangential springs are updated based on the contact modes. The tangential impulse is updated as a function of the accumulating normal impulse. Details of the procedure for such update can be found in Jia (202). The output from each compliance model includes the derivatives of the tangential impulses I u and I w (or, I x and I y ) with respect to the normal impulse I n (or I z ). Denote by e n and e z the coefficients of restitution for the cue-ball and ball-table impacts, respectively. The normal stiffness k n (or k z ) scales up by a factor of /e 2 n (or /e 2 z) every time the corresponding normal impact finishes compression. Its ratio to the tangential stiffness at the contact needs to be scaled up by the same factor. Figure 3 illustrates integration of the models for multiple impacts and compliant impact with dynamics and contact kinematics. We apply the multiple impact model to compute accumulations of the two normal impulses I n and I z. It is responsible for generating the state sequence. The shot by the cue stick has four states: S, during which both the cue-ball and the ball-table impacts are active; S 2, during which only the ball-table impact is active; S 3, during which only the cue-ball impact is active; and S 4, the final state. The shot starts with S and ends with S 4. The state transition diagram has the same structure 2 as that depicted in Figure 4 except for respective 2 The diagram now is even simplified because the coefficients of restitution for both impacts are non-zero (and the contact stiffnesses are finite and non-zero). 33

34 replacements of v v 2 and v 2 with the normal velocity components ṅ and ż. A cue-ball impact is in compression when ṅ < 0. A ball-table impact is in compression when ż < 0. In a state, the primary impulse is the variable that drives the entire system. All tangential impulses are eventually functions of the primary impulse in the state. 8.2 Algorithm for Simulating a Billiard Shot Algorithm simulates a billiard shot via numerical integration. On line 7, update of the strain energies E n and E z stored by the two normal springs proceeds according to the derivative (5) of the strain energy stored by a single impact with respect to impulse, or more specifically here, de n di n = ṅ and de z di z = ż. (55) Line 3 tests if the cue-ball impact is active. If so, lines 4 7 calls upon its compliant impact model to determine the contact mode and evaluate the derivatives of the two tangential impulses at the contact. Lines 2 24 handles the situation where the ball-table impact is active. Lines 7 2, 28 29, and 32 are carried out by the introduced multiple impact model. Below we explain initialization of a state in line 3. The single-impact states S 2 and S 3 need not be initialized, because the impulse derivative di n /di z or di z /di n inherits its value from the previous state S, and so do the strain energies and the active contact modes. State S either is the first state or follows one of S 2 and S 3. If S is the first state, di z /di n has the initial value zero by (29). Let δ be the step size of numerical integration (over either I n or I z ) used in Algorithm. So we determine the value ρ of di z /di n at I n = δ. Use the compliant impact model to determine whether the initial cue-ball contact sticks or slips, and along with it, the derivative di (δ) = ˆn + di u (δ)û + di w (δ)ŵ. di n di n di n Initially, I 2 is dominated by I, and thus ignored in obtaining the ball-table contact velocity using (52). Apply the same initialization approach from Section 4.2: ρ = di z di n (δ) = k z Ez (δ) k n E n (δ), where E n (δ) and E z (δ) are in terms of I (δ) (with I 2 (δ) ignored in comparison) via one step numerical integration over I n and I z, respectively. This will set up a quadratic equation in ρ. Solve it for ρ. We approximate I z (δ) ρi n (δ) = ρδ and use the compliant impact model to obtain I x (δ) and I y (δ), and thus I 2 (δ). If S succeeds S 3, that is, the ball-table impact is reactivated during the cue-ball impact, we can estimate di z /di n similarly as in the above case, except plugging in the values of E n, v, v 2, and di /di n at the transition point If S succeeds S 2, that is, the cue-ball impact is reactivated during the ball-table impact, suppose the transition S 2 S happens at I z = I (0) z ball-table impulse just after S starts, we can solve for di n /di z at I z (0) equation.. By ignoring the ball-ball impulse compared with the + δ from another quadratic 34

35 Algorithm Billiard Shot : state s S 2: while not S 4 do 3: initialization 4: while state s do 5: update cue and ball velocities according to impact dynamics (48) (50) 6: update contact velocities according to contact kinematics (5) (52) 7: update strain energies using I n and I z according to (55) 8: if s = S 3 or s is the initial state or (s = S and previous state is S 3 ) then 9: di z /di n k z E z /k n E n 0: else : di n /di z k n E n /k z E z 2: end if 3: if s S 2 then 4: update strain energies stored by the u- and w-springs 5: determine cue-ball contact mode 6: evaluate di u /di n and di w /di n 7: update I u and I w via one-step integration 8: increment I n 9: end if 20: if s S 3 then 2: update strain energies stored by the x and y-springs 22: determine ball-table contact mode 23: evaluate di x /di z and di y /di z 24: update I x and I y via one-step integration 25: increment I z 26: end if 27: update I and I 2 according to (53) 28: if any active impact ends compression or restitution then 29: update impact phase 30: end if 3: end while 32: s next state according to the transition diagram in Figure 4 33: end while 35

36 9 Experiments In this section, we will first validate the ball-ball-table collision model described in Sections 3 7 using an experiment with two ping pong balls. Then we will compare measurements from a real massé shot with predictions by Algorithm that integrates the multiple impact model and the compliant impact model. 9. Collision of Ping Pong Balls Against Plexiglass Surface As shown in Figure 4(a), a ping pong ball B was dropped by hand onto another one B 2 resting on a plexiglass block. The two balls were identical with mass kg and radius 0.09m. The block was placed horizontally on the marker tray of a (vertical) office whiteboard, and against a vertical axis l drawn on the board. Both balls were positioned almost in contact with the whiteboard so the line through their centers was as close to be parallel to l as possible. To measure the coefficient e 2 of restitution between a ball and the plexiglass surface, we dropped the ball from certain height h onto the surface and recorded the rebounding height h 2 (on the axis by human vision). So e 2 = h 2 /h after ignoring air resistance. Sixteen measurements from different drop heights generated a mean estimate of with a standard deviation of To measure the coefficient e of ball-ball restitution, both balls were hanged by parallel threads at the same height (like two pendula) and a distance of twice the radius apart so the balls are in contact. Next, one ball was raised to a height of h 3 relative to the other one while its attached thread was kept straight. 22 Then it was released, colliding with the other ball, which reached a height of h 4. We obtained e = h 4 /h 3. The mean value of e calculated over eleven different dropping heights was with standard deviation We could not get a good measure of the stiffness k between the two ping pongs or the stiffness k 2 between the ping pong and the plexiglass. We know that the ratio k 2 /k > since the plexiglass is considerably stiffer. A collision trial involved dropping B from a fixed height onto B 2 multiple times, and choosing the one with the highest rebounds of both balls 23. The velocity v 0 of B before the collision and the ball velocities v and v 2 after the collision were calculated from the heights reached by the balls. Results from ten trials, each with a different dropping height, are plotted in Figure 4(b) as pairs of points ( v 0,i,v,i ) and ( v 0,i,v 2,i ), 0 i 9 for the guessed value k 2 /k = 2.25, where v 0 varies from v 0,0 = m/s to v 0,9 = m/s. Also shown are two lines of v and v 2 predicted by the multiple impact model as v 0 varies from 2m/s to 3.6m/s. (The ratios v /v 0 and v 2 /v 0 are invariants to v 0 under Theorem 5.) In Figure 4(c), the same data points are plotted against model predictions with k 2 /k = 2.7. The model generated the same state sequence S,S 3,S 4 in all trials. In (b), the model predictions had the best match with the data over v ; while in (c), the model predictions did over v 2 (except for the two outliers v 2,0 and v 2,9 ). Overall, (c) represents a better match. Despite the rather primitive setup and measurements, the results suggest a reasonably good match between our model and the physical collision process. 22 For more accuracy, the horizontal distance of each ball at its maximum height to its resting position was measured, and the height was calculated from this distance. 23 thus the balls were the closest to be vertically aligned before the collision. 36

37 (m/s) 3.2 (m/s) v 2.8 v v,0.6 v,0.2 v 2.2 v v 2,0 2.0 v 0 (m/s) v 2,0 2.0 v 0 (m/s) l (a) (b) (c) Figure 4: (a) Collision between two ping pong balls and a plexiglass surface (from a video capture); (b) results from 0 trials (dots) vs. predictions (lines) by the impact model (with guessed value k 2 /k = 2.25); and (c) match with k 2 /k = A Massé Billiard Shot To experimentally validate Algorithm, we designed a (manual) mechanical cue (see Figure 5(a)). It was constrained to linear motions by ball bearings embedded inside an aluminum block. A small flat aluminum piece attached to the other end of the cue stick served as a trigger which, connected to the aluminum block via two identical springs, could be pulled and then released to execute a shot on the cue ball. The block rested on an adjustable incline so the cue stick can hit the ball from a wide range of directions. The cue together with the trigger weighed m c = 0.508kg. The cue ball had radius R = m and weight m = 0.070kg. The two springs 24 had stiffness k s = N/m and initial stretches of x 0 = 0.0m. A ruler was fastened to the back side of the aluminum block to measure their 2k s ( x 2 x 2 0 )/m c. changes in length x after the pulling. The cue hit the ball at speed The coefficient of restitution e 2 for a ball-table impact was measured by dropping a ball onto the table and recording its drop height and bounce height, similarly to the measurement method in Section 9. for a ping pong ball. The average value calculated from drops at four different heights was e 2 = The coefficient of restitution e between the cue and the ball was measured in a similar way by dropping the ball onto an identical cue tip resting bottom-up on a hard surface. Our estimate was We taped four object balls (of the same material as the cue ball) from the top so they would slide together. Then tilted the pool table until the balls started sliding. The coefficient µ s of sliding friction was calculated as the slope of the tilted table (µ s = ). We let the cue ball roll down a slope of height h set up on the pool table, and measured how far it would continue on the table. The ball velocity right at the bottom of the slope was calculated from the height of the slope. Then the coefficient µ r of rolling friction was estimated from the rolling distance d on the table due to the deceleration of 5 7 µ rg under friction, where g > 0 is 24 model no. 9564K376 from McMaster-Carr, Inc. 37

38 incline adjustment ball bearings inside trigger spring y x θ φ cue tip (a) (b) (c) Figure 5: (a) Adjustable billiard shooter using springs and ball bearings (embedded). Setup for a massé shot: (b) side view; and (c) top-down view. the gravitational acceleration. 25 It is easy to derive that µ r = 7h/(5d). The calculated value was The coefficient of friction between the cue and the ball can be as high as 0.7 depending on the amount of chalk (Cross). We used the value 0.7 in the experiment. The relative stiffnesses were set as k 2 /k = 0.8, η 2 = 7, and η2 2 = 0. Here, k 2/k is the ratio between the normal stiffnesses at the ball-table and cue-ball contacts, η 2 and η2 2 the ratios between the normal and tangential stiffnesses at the cue-ball and ball-table contacts, respectively. Unable to measure these three parameters, we had to guess their values. The felt was softer than the cue tip but not by too much because of the hard surface of the table underneath it. So we chose k 2 /k to be less than one but not by too much. The cue-ball contact likely had less tangential compliance than the ball-table contact, so we set η < η 2. A massé shot was performed. Figure 5 (b) and (c) present two different views of the pre-shot configuration. The cue stick tilted at an angle of φ = 2 5π with the pool table. Its projection onto the table formed an angle of θ = with the negative x-axis which was aligned with a table edge. The cue tip hit at a location on the cue ball with measured outward normal n = (0.4805, , 0.873). Before the shot, the springs were stretched by x = 0.079m due to the hand pull. Based on the mechanical model for the shooter, we calculated that the cue tip hit the ball at the speed v c = m/s. After the massé shot, the cue ball slid along a parabolic trajectory before rolling along a straight trajectory to halt. A video of the shot was taken by a Cannon PowerShot SD790S digital camera at 30 frames/s. The video was sequenced at the same frame rate, and in each frame, the ball s center was measured manually. Figure 6(a) plots the positions (dots) of the ball s center every /30th of a second during the motion. Also shown are the trajectory fit over these positions along with the parabola, and the line on which the ball had moved. Note that the leftmost dot represents only the last position before the ball exited the camera s field of view. In Jia et al. (20b), an involved analysis is presented for the trajectory of a billiard given its initial velocity and angular velocity, 25 The deceleration of a rolling ball is derived as equation (6) in Jia et al. (20b). 38